How do you use Blender to make Pokemon attacks like hyper beam and solar beam?
Never tried to create any effect like that, but it is possible that particles are not the most effective way to simulate those beams.

Have you tried looking into create "Plasma Wave" effects like these using geometry instead? Like Abstract Waves or Plasma Waves or any other

Perhaps actually modeling it with geometry might be easier. Adding some effects over it, like added particles, volumetric materials, or even in post production, like glow and blur may improve the final aspect.